Compact objects. Neutron stars: matter under extreme conditions. General properties First stimation of size and mass. Equation of state of -stable matter TOV equations and mass, radius, and density profiles. Maximum mass. “Compact objects for everyone: a real experiment”, J. Piekarewicz et al. Eur. J. Phys. 26 (2005) 695-709 “ Neutron stars for undergraduates “, R. Silbar, S. Reddy, Am. J. Phys. 72, 892 (2004). TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

What is a neutron star? The answer depends to whom you ask. Very multidisciplanary subject For astronomers they are very small stars visible as radio-pulsars or also as a source of X and  rays. For particle physicists they are a source of neutrinos (in some early stage of its evolution) and one of the few places were quark matter can exist. For nuclear physicist , they are consider as the largest nucleus in the univers For cosmologists, they are considered as almost black-holes. For computational physicists, the simulation of their birth and evolution is a real headache. In any case they offer fascinating phenomena under extreme density conditions. Their properties are sensitive to the equation of state of neutron matter in a wide range of densities! A very active field of research!

A neutron star can be considered as gigantic nucleus, with a mean density of the order of  1014 – 10 15 g cm-3. A neutron star contains  1057 nucleons, 90% being neutrons. They are the densest macroscopic objects in the Universe They are about 10 km diameter, and about 1-2 MS, with and average density 1014 – 10 15 g cm-3 to be compared with the average density in the universe  10 -30 g cm-3 , the density of the Earth 5.5 g cm-3 and is of the same order of the density inside nuclei 2.8 1014 g cm-3 Because of the small size and the high density a neutron star has a surface gravitational field of about 3 105 times that of the Earth and a very large gravitational energy for a star with M= 1.4 MS and R= 10 km.

N  10 57 neutrons M  1 MS

Now a days we can explore the phase diagram of hadronic matter. There are several experimental facilities (RHIC and FAIR in the future) to explore different regions of the phase diagram. The observational data and study of neutron stars offers a unique possibility to understand hadronic matter at very high density.

Size comparison of a neutron star with a characteristic white dwarf And the earth. Characteristic scape velocities for 1 solar mass, white dwarf with a radius of 5000 km and a neutron star with 10 km radius: Sun 600 Km/s White dwarf 7000 km/s Neutron star 160000 km/s

Liquid drop model picture sphere of constant density with only neutrons the energy described by the mass phormula, modified with a gravitational energy term For Z=0, Adding the gravitaional energy : Whre mn is the mass of the neutron, we get

If N is large we can neglect the term N2/3 av= 16 MeV, asym= 30 MeV, G= 6.67 10-39 197.3 MeV fm (c4/GeV2) mn= 0.939 GeV/c2 r0= 1.15 fm If N is large we can neglect the term N2/3 Gravitation helps to bind the system and we can determine the minimun number of neutrons to bind the neutron star, is given B(0,Nmin )  0 R is fixed by the number of neutrons!!!!

N  1056 – 10 57 , M  1 MS and R  10 km And a density ,   1014 – 10 15 g/cm3 A neutron star looks as gigantic nucleus where the neutrons are bound by gravity, and the collapse is avoided by the nuclear forces

All signals from the neutron star go through the surface  it is crucial to have a good knowledge of the crust!!

One can use this EoS to study neutron and nuclear matter.

- stable neutron star matter Neutron stars are not made only of neutrons but also include a small fraction of protons ans electron! The presence of lectrons guarantees charge neutrality. There should be chemical equilibrium respect to the weak interaction  as many neutron decays as electron capture reactions The equilibrium can be expressed in terms of the chemical potentials. Now since the mean free path of the neutrinos in a neutron star is larger than the typical radius of such star, we assume that the neutrino escape freely from the neutron star, and do not play a role in the equilibrium for these reactions.

n = p + e - equilibrium conditions in terms of the chemical potentials. ρp = ρe A possible equivalent way to find the composition of -stable matter consists in minimizing the total energy density of the system constrained by the subsidiary conditions that express the conservation of the different charges. In the case of neutron star we have two conserved charges: total barionic density and total electric charge!. For the case that we consider, with p,n, and e- we have: Minization conditions We have two Lagrange multipliers and we should minimize respect to them

In general there will be as many independent chemical potentials as conserved charges! Remember

It is interesting to realice that the relations between the chemical potentials is defined by the weak interaction. However, the chemical potentials of the hadrons are determined by the strong interaction. We can also include muons! It could be that the chemical potential of electrons increases rather fast and becomes equal to the bare mass of the muons. Than the minimization of F says that the system will prefere to start to populate the Fermi sea of muons. We still have only two independent chemical potentials And the minimization produces the follwoing equations

The chemical potential of the electron in -stable matter is  100 MeV Once the rest mass of the muon is exceeded, the electrons will start to decay through a weak process, that would not take place In the free space. Composition of -stable neutron star matter. On the left panel we have only neutrons,protons and electrons. In the right one we include also muons.

As soon as the chemical potential of the neutrons becomes sufficiently large. Energetic neutrons can decay via strangeness non-conserving weak interactions into  hyperons leading to a  Fermi sea with  =n .

M() = 1115 MeV M(-) = 1197 MeV One can also expect that - can appear through e- + n  - + e at lower densities than than that of the , even though the - is more massive. The reason being, that the above process removes both an energetic neutron and an energetic electron. The negative charged hyperons appear in the ground state of -stable matter when its mass equals n + e .

Composition of the -stable neutron star matter. The composition depends on the model used for the interaction. The dashed lines correspond to the NO inclusion of YY interactions. In principle should nof affect the onset density of - .

Each curve correspond to a different composition. Leptonic contributions are included! Treated as relativistic Fermi seas. The presence of the hyperons produces a softening of the EoS.

M(r) is the gravitational mass enclosed in a radius r, and (r) is the mass-energy density. They contain the bare mass and take care of the binding energy. In the limit of c , one recovers Newton’s experssion. The factors in the numerator represent special relativity corrections of order v2/c2 , and the factor in the denominator is a general relativistic correction. All factors tend to 1 in the non-relativistic limit. All corrections are positive defined, it is as if Newtonian gravity becomes stronger for any value of r.

NN 1.89 2.11 10.3 NN-YN 1.47 1.60 10.2 NN-YN-YY 1.34 1.44 10.2 The radius does not change so much. The maximum mass is probably too small. The different EoS produce different maximum masses. When the hyperons are present the maximum mass decreases. If rotation of the star is taken into account, the maximum mass increases. It is as if the centrifugal effects increase the pressure or that you need more mass to compensate for the centrifugal effects.

Composition as a function of density (upper panel) and composition inside T he star. The onset density of the hyperons allows to define an hyperonic radius